local thermal stress factor of pipe-nozzle · 2020. 5. 12. · bulletin 107 method in pipe-nozzle...
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New Jersey Institute of TechnologyDigital Commons @ NJIT
Dissertations Theses and Dissertations
Fall 1994
Local thermal stress factor of pipe-nozzleDavid Chihwei ChenNew Jersey Institute of Technology
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Order N um ber 9514441
L ocal th erm al stress fa c to r o f p ip e-n ozzle
Chen, D avid Chihwei, Ph.D .
New Jersey Institute of Technology, 1994
C opyright © 1994 by C hen, D avid Chihw ei. A ll righ ts reserved.
U M I300 N. Zeeb Rd.Ann Arbor. MI 48106
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ABSTRACT
LOCAL THERMAL STRESS FACTOR OF PIPE-NOZZLE
byDavid Chihwei Chen
A comprehensive study o f local thermal stresses at the juncture o f pipe -nozzle is
presented in this thesis. The thermal loading is assumed to be a linear thermal gradient
across the thickness o f the pipe and nozzle. Currently, there exists neither experimental
nor analytical data that is sufficient for pressure vessel designers to analyze the local
thermal stresses at the juncture o f pipe-nozzle. In order to provide a comprehensive
database to calculate these thermal stresses, the finite element technique is used to provide
a series o f local thermal stress factor plots as a function o f pipe-nozzle geometrical
parameters.
For the local thermal stresses on the juncture o f pipe-nozzle, the longitudinal and
circumferential thermal stress factors due to the thermal loading are presented in a series
o f plots as functions o f gamma, y (pipe mean radis/pipe thickness) and beta, P (nozzle
mean radius/pipe radius). The gamma values vary from 10 to 300 and beta values vary
from 0.1 to 1.0. These stress factors would complement the welding Research Council
Bulletin 107 method in pipe-nozzle stress analysis which did not include the effect o f local
thermal stresses.
To ensure the convergence o f the finite element results, two major parameters
were thoroughly studied. First, to minimize the influence o f boundary conditions on the
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thermal stresses around the juncture o f the pipe-nozzle, the geometrical parameter alphap,
otp, (pipe length/pipe mean radius) is found to be at least equal to 8.0 as well as alpha,,,
oc„, (nozzle length/nozzle mean radius) at least to be 4.0. Next, 96 node points must be
assigned at the juncture o f pipe-nozzle. As a result, approximately 5000 node points and
3000 plus elements were needed for the computation. Numerical examples are also
presented in this thesis to demonstrate how the thermal stress components complement the
WRC 107 local stress computation due to external loadings.
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LOCAL THERMAL STRESS FACTOR OF PIPE-NOZZLE
byDavid Chihwei Chen
A Dissertation Submitted to the Faculty of
New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
Department of Mechanical and Industrial Engineering
O ctober 1994
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Copyright © 1994 by David Chihwei Chen
ALL RIGHTS RESERVED
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APPROVAL PAGE
LOCAL THERMAL STRESS FACTOR OF PIPE-NOZZLE
David Chihwei Chen
Dr. Benedict C. Sun, Dissertation Advisor Date Associate Professor of Engineering Technology, NJIT
Dr. Rong-Yaw Chen, Committee Chair Date Professor of Mechanical Engineering and Associate Chairperson an Graduate Advisor of Mechanical Engineerin JI /
Dr. Bernard Koplik, Committee Member Date Professor of Mechanical Engineering and Chairperson of the Department of Mechanical and Industrial Engineering, NJIT
Dr. Nouri Levy, Committee Member Date Associate Professor of echanical Engineering, NJIT
Dr. C.T. Thomas Hsu, Committee Member Date Professor of Civil and Environmental Engineering, NJIT
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BIOGRAPHICAL SKETCH
Author: David Chihwei Chen
Degree: Doctor of Philosophy in Mechanical and Industrial Engineering
Date: October 1994
Undergraduate and Graduate Education:
• Doctor of Philosophy in Mechanical and Industrial Engineering , New Jersey Institute Technology, Newark, New Jersey, 1994
• Master of Science in Mechanical Engineering , New Jersey Institute Technology, Newark, New Jersey, 1987
• Bachelor of Science in Mechanical Engineering , Tamkang University, Tamsui, Taiwan, 1984
Major: Mechanical Engineering
Presentations and Publications:
David C. Chen "On Longitudinal and Transverse Shear Spring Constants at the Juncture of Piping-nozzle." Master Thesis, NJIT Newark, New Jersey, May 1987
Position Held:
Project Manager Airco Gases, Engineering System Dept. D.I.E.T. 575 Mountain Ave., Murray Hill, New Jersey 07974
Professional License and Memberships:
Professional Engineer License, Pennsylvania, 1993 Member of American Society of Mechanical Engineers Member of National Association of Corrosion Engineers
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This dissertation is dedicated to
my parents and all my family members
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ACKNOWLEDGMENT
The author wishes to express his sincere appreciation to his Dissertation advisor, Dr.
Benedict Sun, for his guidance, friendship, and moral support throughout this research.
Special thanks to Dr. Rong-Yaw Chen, Dr. Bernard Koplik, Dr. Nouri Levy, and
Dr. C. T. Thomas Hsu for serving as members o f the committee and their kindly
suggestions and support.
Also, the author would like to express his gratitude to his family for their love,
understanding and support to the achievement o f this dissertation.
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TABLE OF CONTENTS
Chapter Page
1 INTRODUCTION ............................................................................................................ 1
2 LITERATURE SURVEY ................................................................................................. 4
3 BASIC THEORY ............................................................................................................. 12
3.1 Derivation o f Equations for Deflections Due to the Thermal Loading . . . . 12
3.2 Thermal Stress Factors ....................................................................................... 20
4 FINITE ELEMENT MODEL ......................................................................................... 24
4.1 General ................................................................................................................... 24
4.2 Assumption ............................................................................................................ 25
4.3 Asymptotic Studies ............................................................................................... 25
4.3.1 Asymptotic Study o f Node Points at Juncture o f Pipe-nozzle ........ 25
4.3.2 Asymptotic Study o f the a and a n ........................................................ 26
4.4 Normalization Studies ......................................................................................... 26
4.4.1 Case I, II, III ............................................................................................... 26
4.4.2 Case IV ........................................................................................................ 27
5 COMPARISON OF DATA ............................................................................................. 28
5.1 General ................................................................................................................... 28
5.2 Comparison o f Thermal Stress Factors ............................................................ 28
5.2.1 Case 1 ............................................................................................................ 28
5.2.2 Case 2 ............................................................................................................ 30
5.3 Comparison o f Thermal Stresses ....................................................................... 32
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Chapter Page
6 NUMERICAL EXAMPLES ........................................................................................... 34
6.1 Example I ............................................................................................................... 34
6.2 Example II .............................................................................................................. 38
7 CONCLUSIONS ............................................................................................................... 42
APPENDIX A THERMAL STRESS FACTOR PLOTS ........................................ 43
APPENDIX B ASYMPTOTIC STUDY OF NODE POINTS ATJUNCTURE OF PIPE-NOZZLE ....................................................... 60
APPENDIX C ASYMPTOTIC STUDY OF a p AND a N ...................................... 77
APPENDIX D NORMALIZATION STUDIES ........................................................ 110
APPENDIX E COMPARISON OF DATA - CASE 2 ............................................. 123
REFERENCES ........................................................................................................................132
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LIST OF TABLES
Table Page
1 Modified Stress Computation Table o f WRC 107 Including Local ThermalStresses ............................................................................................................................ 22
2 List o f Thermal Stresses and Thermal Stress Factors Given in the Case 2 ........ 31
3 Comparison o f Local Thermal Stresses at Pipe-nozzle and Theoretical ThermalStresses at Regular Long Hollow Pipe ...................................................................... 33
4 Material Properties o f the Illustrating Pipe-nozzle Model ...................................... 35
5 Geometrical Parameters and Dimensions o f Example for Calculation o f LocalStresses on the Pipe o f Pipe-nozzle Model .............................................................. 35
6 Computation Table o f Thermal Stress Factors on the Pipe .................................... 36
7 Modified Stress Computation Table o f WRC 107 Including Local ThermalStress on the Pipe - Numerical Example .................................................................... 37
8 Geometrical Parameters and Dimensions o f Example for Calculation o f LocalStresses on Nozzle o f Pipe-nozzle Model ................................................................ 39
9 Computation Table o f the Thermal Stress Factors on the Nozzle ....................... 39
10 Computation Table o f Example for Calculation o f Local Stresses on Nozzle o fPipe-nozzle Model ........................................................................................................ 40
11 Local Stress Factors on the Nozzle from Lin [45] ................................................... 41
D -l Material Properties, Geometric Parameters and Dimensions o f Case # 1and Case # 2 .................................................................................................................... I l l
D-2 Thermal Stresses & Stress Factors Comparison Table at Node Point Ao f Case # 1 and Case # 2 ..............................................................................................112
D-3 Thermal Stresses & Stress Factors Comparison Table at Node Point Co f Case # 1 and Case # 2 .............................................................................................. 113
D-4 Material Properties, Geometric Parameters and Dimensions o f Case # 3and Case # 4 .................................................................................................................... 114
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Table Page
D-5 Thermal Stresses & Stress Factors Comparison Table at N ode Point Ao f Case # 3 and Case # 4 ................................................................................................ 115
D -6 Thermal Stresses & Stress Factors Comparison Table at N ode Point Co f Case # 3 and Case # 4 ................................................................................................ 116
D-7 M aterial Properties, Geometric Parameters and Dimensions o f Case # 5and Case # 6 ....................................................................................................................... 117
D-8 Thermal Stresses & Stress Factors Comparison Table at N ode Point Ao f Case # 5 and Case # 6 ................................................................................................ 118
D -9 Thermal Stresses & Stress Factors Comparison Table at N ode Point Co f Case # 5 and Case # 6 ................................................................................................ 119
D -10 M aterial Properties, Geometric Parameters and Dimensions o f Case # 7and Case # 8 ....................................................................................................................... 120
D -l 1 Thermal Stresses & Stress Factors Comparison Table at N ode Point Ao f Case # 7 and Case # 8 ................................................................................................ 121
D - l2 Thermal Stresses & Stress Factors Comparison Table at N ode Point Co f Case # 7 and Case # 8 ................................................................................................ 122
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LIST OF FIGURES
Figure Page
1 Typical Configuration o f Pipe-nozzle Juncture under AxisymmetricalTemperature Distribution ............................................................................................. 3
2 Literature Survey Table ............................................................................................. 7
3 Cylindrical Coordinate Applied to a Cylindrical Pipe with DisplacementU, V, and W in X, (p, and Z Direction Respectively ............................................ 19
4 Different Loadings Applied on the Juncture o f Pipe-nozzle ................................. 23
5 Data Comparison o f Case 1 - Thermal Stress Factor in the LongitudinalDirection at Point Au o f the Pipe ............................................................................... 29
1T Thermal Stress Factor in the Longitudinal Direction at Point A u o f the Pipe . . . 44
2T Thermal Stress Factor in the Longitudinal Direction at Point AL o f the Pipe . . . 45
3T Thermal Stress Factor in the Longitudinal Direction at Point Cy o f the Pipe . . . 46
4T Thermal Stress Factor in the Longitudinal Direction at Point CL o f the Pipe .. 47
5T Thermal Stress Factor in the Circumferential Direction at Point Ay o f thePipe ......................................................................................................................................48
6T Thermal Stress Factor in the Circumferential Direction at Point AL o f thePipe .................................................................................................................................... 49
7T Thermal Stress Factor in the Circumferential Direction at Point C,j o f thePipe .................................................................................................................................. 50
8T Thermal Stress Factor in the Circumferential Direction at Point CL o f thePipe .................................................................................................................................. 51
9T Thermal Stress Factor in the Longitudinal Direction at Point A0 o f theNozzle .............................................................................................................................. 52
10T Thermal Stress Factor in the Longitudinal Direction at Point A, o f theNozzle .............................................................................................................................. 53
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Figure Page
11T Thermal Stress Factor in the Longitudinal Direction at Point C0 o f theNozzle .............................................................................................................................. 54
12T Thermal Stress Factor in the Longitudinal Direction at Point C, o f theNozzle .............................................................................................................................. 55
13T Thermal Stress Factor in the Circumferential Direction at Point A0 o f theNozzle .............................................................................................................................. 56
14T Thermal Stress Factor in the Circumferential Direction at Point A, o f theNozzle .............................................................................................................................. 57
15T Thermal Stress Factor in the Circumferential Direction at Point CQ o f theNozzle .............................................................................................................................. 58
16T Thermal Stress Factor in the Circumferential Direction at Point C, o f theNozzle .............................................................................................................................. 59
B 1 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point Au o f the P ip e ....................................................... 61
B2 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point AL o f the Pipe ....................................................... 62
B3 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point Cy o f the Pipe ....................................................... 63
B4 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point CL o f the Pipe ....................................................... 64
B5 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point Au o f the Pipe ....................................................... 65
B6 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point AL o f the Pipe ....................................................... 66
B7 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point Cy o f the Pipe ....................................................... 67
B8 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point CL o f the Pipe ....................................................... 68
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Figure Page
B9 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point A0 o f the Nozzle .................................................. 69
BIO Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point A; o f the N o z z le ................................................... 70
B 11 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point C0 o f the Nozzle ................................................. 71
B12 Convergence o f Node Points at Juncture o f Pipe-nozzle for LongitudinalThermal Stress Factors at Point Cj o f the N o z z le ................................................... 72
B13 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point A0 o f the Nozzle ................................................. 73
B14 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point A, o f the N o z z le ................................................... 74
B15 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point CQ o f the Nozzle ................................................. 75
B16 Convergence o f Node Points at Juncture o f Pipe-nozzle for CircumferentialThermal Stress Factors at Point C, o f the Nozzle ................................................... 76
C l Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point Ay o f the P ip e ....................................................... 78
C2 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point AL o f the Pipe ....................................................... 79
C3 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point Cu o f the Pipe ....................................................... 80
C4 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point CL o f the Pipe ....................................................... 81
C5 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point Ay o f the Pipe ....................................................... 82
C6 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point AL o f the Pipe ....................................................... 83
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Figure Page
C7 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point Cu o f the Pipe ....................................................... 84
C8 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point CL o f the Pipe ....................................................... 85
C9 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point Ao o f the Nozzle .................................................. 86
CIO Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point A, o f the N o z z le ................................................... 87
C l 1 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point C0 o f the Nozzle .................................................. 88
C12 Percentage o f Improvement o f Larger a p to Previous a p for LongitudinalThermal Stress Factors at Point C, o f the Nozzle ................................................... 89
C13 Percentage o f Improvement o f Larger a t o Previous a p for CircumferentialThermal Stress Factors at Point A0 o f the Nozzle .................................................. 90
C14 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point A, o f the N o z z le .................................................... 91
C l 5 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point C0 o f the Nozzle .................................................. 92
C l 6 Percentage o f Improvement o f Larger a p to Previous a p for CircumferentialThermal Stress Factors at Point C, o f the Nozzle .................................................... 93
C l 7 Percentage o f Improvement o f Larger a n to Previous a nfor LongitudinalThermal Stress Factors at Point Au o f the Pipe ....................................................... 94
Cl 8 Percentage o f Improvement o f Larger a n to Previous a nfor LongitudinalThermal Stress Factors at Point AL o f the Pipe ....................................................... 95
C l 9 Percentage o f Improvement o f Larger a n to Previous a n for LongitudinalThermal Stress Factors at Point Cu o f the Pipe ....................................................... 96
C20 Percentage o f Improvement o f Larger a n to Previous a n for LongitudinalThermal Stress Factors at Point CL o f the Pipe ....................................................... 97
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Figure Page
C21 Percentage o f Improvement o f Larger a n to Previous a nfor CircumferentialThermal Stress Factors at Point Al, o f the Pipe ......................................................... 98
C22 Percentage o f Improvement o f Larger a n to Previous a nfor CircumferentialThermal Stress Factors at Point AL o f the Pipe ......................................................... 99
C23 Percentage o f Improvement o f Larger a n to Previous a n for CircumferentialThermal Stress Factors at Point Cy o f the Pipe ......................................................... 100
C24 Percentage o f Improvement o f Larger a n to Previous a n for CircumferentialThermal Stress Factors at Point CL o f the Pipe ......................................................... 101
C25 Percentage o f Improvement o f Larger a n to Previous a nfor LongitudinalThermal Stress Factors at Point A0 o f the N ozzle ...................................................102
C26 Percentage o f Improvement o f Larger a n to Previous a n for LongitudinalThermal Stress Factors at Point \ o f the Nozzle ......................................................103
C27 Percentage o f Improvement o f Larger a n to Previous a n for LongitudinalThermal Stress Factors at Point CD o f the N ozzle ...................................................104
C28 Percentage o f Improvement o f Larger a n to Previous a n for LongitudinalThermal Stress Factors at Point C ( o f the Nozzle ......................................................105
C29 Percentage o f Improvement o f Larger a n to Previous a nfor CircumferentialThermal Stress Factors at Point A0 o f the N ozzle ....................................................106
C30 Percentage o f Improvement o f Larger a n to Previous a nfor CircumferentialThermal Stress Factors at Point Aj o f the Nozzle ..................................................... 107
C 3 1 Percentage o f Improvement o f Larger a n to Previous a n for CircumferentialThermal Stress Factors at Point CQ o f the Nozzle ...................................................108
C32 Percentage o f Improvement o f Larger a n to Previous a n for CircumferentialThermal Stress Factors at Point C, o f the Nozzle ......................................................109
E l D ata Comparison o f Case 2 - Thermal Stress Factor in the LongitudinalD irection at Point A0 o f the Nozzle ..............................................................................124
E2 D ata Comparison o f Case 2 - Thermal Stress Factor in the LongitudinalDirection at Point A; o f the Nozzle .............................................................................. 125
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Figure Page
E3 Data Comparison o f Case 2 - Thermal Stress Factor in the LongitudinalDirection at Point C0 o f the Nozzle ..............................................................................126
E4 Data Comparison o f Case 2 - Thermal Stress Factor in the LongitudinalDirection at Point Cj o f the Nozzle ..............................................................................127
E5 Data Comparison o f Case 2 - Thermal Stress Factor in the CircumferentialDirection at Point A0 o f the Nozzle ..............................................................................128
E6 Data Comparison o f Case 2 - Thermal Stress Factor in the CircumferentialDirection at Point A, o f the Nozzle ..............................................................................129
E7 Data Comparison o f Case 2 - Thermal Stress Factor in the CircumferentialDirection at Point C0 o f the Nozzle ..............................................................................130
E8 Data Comparison o f Case 2 - Thermal Stress Factor in the CircumferentialDirection at Point C, o f the Nozzle ..............................................................................131
xvi
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NOMENCLATURES
pipe Length / pipe mean radius
nozzle Length / nozzle mean radius
coefficient o f thermal expansion
nozzle mean radius / pipe mean Radius
pipe mean radius / pipe thickness
Poisson's ratio
external heat transfer coefficient, ft-lb / hr-in2-
internal heat transfer coefficient, ft-lb / hr-in2-
local thermal stress, psi
Young's Modulus, psi
pipe thickness / 2 , in
nozzle thickness / 2 , in
local thermal stress factor
pipe length, in
nozzle length, in
shell moment resultants, in-lb
thermal moment, in-lb
shell force resultants, lb
thermal membrane force, lb
pipe mean radius, in
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= pipe outside radius, in
tP - pipe thickness, in
rm = nozzle mean radius, in
rn = nozzle outside radius, in
tn = nozzle thickness, in
T, = internal temperature, °F
T0 = external temperature, °F
Tnm = mean temperature at the nozzle, °F
Tpm - mean temperature at the pipe, °F
Tnd = normal temperature different at the nozzle, ‘
T1Pd = normal temperature different at the pipe, °F
u = displacement in x direction at the nozzle, in
U = displacement in x direction at the pipe, in
V = displacement in
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CHAPTER 1
INTRODUCTION
Thermal stresses analysis at the juncture o f pipe-nozzle is one o f the critical factor for
pressure vessel design. From linear thin-shell theory, an analytical solution based on
Morley's equations, which has nearly the same simple form as the well-known Donnell
equations, had been presented by D. H, Van Campen [1] . To date, only a few special
cases were reported based on either experimental or analytical techniques. However, these
available literatures and publications are so limited that they are not sufficient to be used
as a design guide for most o f the pipe-nozzle stress analysis. In order to provide a
comprehensive database for thermal stresses on pipe-nozzle, the finite element analysis
method is used in this thesis based on the assumption that the nozzle thickness is
proportional to the pipe thickness by beta (tn = P tp). It covered the following studies:
1. The data ranges o f the geometrical parameters, beta, P (nozzle mean radius/pipe
mean radius) are from 0.1 to 1.0, and gamma, y (pipe mean radius/pipe, thickness) are
from 10 to 300.
2. For the accuracy o f the results, independent o f the boundary conditions, the
geometrical parameters, alpha p, a p (pipe length/pipe mean radius) is at least equal to 8.0,
and alphan, a n (nozzle length/nozzle mean radius) is at least equal to 4.0.
3. For the optimization study o f node point number at the pipe-nozzle juncture, 96
node points are required at the juncture o f the pipe-nozzle lull model.
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4. The resulting thermal stresses on both the pipe and the nozzle around the pipe-nozzle
juncture are normalized as thermal stress factors and presented in a series o f sixteen plots
as function o f P and y. These plots cover the membrane and bending stresses in
longitudinal and circumferential directions on both the inside and the outside surfaces o f
the pipe, as well as the nozzle, at point A and C on X-Z and Y-Z planes, respectively. A
typical configuration o f the pipe-nozzle is shown in Figure 1.
These local thermal stress may be used in conjunction with local stresses from other
external loadings, such as radial load, circumferential moment and longitudinal moment as
well as shear stresses induced by shear forces and torsional moment, which had been
published by Welding Research Council Bulletin 107 [2],
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^ S E E DETA IL ‘A 1
AuP IP E
DETAIL 'A '
Figure 1 Typical configuration o f pipe-nozzle juncture under axisymmetrical temperature distribution
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CHAPTER 2
LITERATURE SURVEY
Since the middle o f 1960's, some studies on local stresses around pipe-nozzle juncture
using theoretical [3][4][5] and experimental [6][7][8][9] analyses due to mechanical and
thermal loadings had been published. In 1968, Manschot [10] presented a numerical
computing method for thermal stresses in thin-walled, tee-type cylinder. In 1969, Van
Campen [1] introduced a solution o f the Morley partial differential shell equation and
numerical method to calculate the local thermal stress o f an equal size tee ( p = 1 ), and
Cranch, et. al., [11] had an investigation on thermal stresses o f circular pipe attached to a
spherical shell and provided some normalized thermal stress factor plots with geometrical
parameter beta, P, (attached cylinder pipe mean radius/ spherical shell mean radius) is
equal to 0.03 , and gamma , y, (shell mean radius/shell thickness) is equal to 169.
After the 1970s, the finite element method had been applied by some researchers
[12], Also, the large computer and the Finite Element Analysis, FEA, code had been
employed to analyze the thermal stresses around the cylinder-to-cylinder juncture [13], In
the meantime, quite a few o f the revised theoretical and experimental [16][17] studies on
the same topic had been published. Van Campen, et. al. [14] in 1972 and Fullard [15] in
1973, both presented the local thermal stresses on the intersection o f small diameter ratio
o f nozzle-to-shell with P less than 0.4. In 1977, Cesari [18] developed a 2-D equivalent
nozzle-cyiinder model to study the local thermal stresses on the juncture o f
nozzle-to-cylinder. In his case study, a special case with P = 0.12, and y = 28.57
4
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(vessel radius = 191.4 mm, vessel thickness = 6.7 mm and nozzle radius = 24.675 mm,
nozzle thickness = 1.35 mm) had been analyzed. Independently also, Gantayst, et.al.
[19] in 1977, presented a finite element procedure and the associated programs for the
analysis o f thin and thick walled tubular tee joint under thermal loading with beta o f 0.5,
and gamma o f 100.
In the similar study field, a conical nozzle on spherical shell had been published by
Jayaraman, et. al. [20], Meanwhile, transient thermal stresses on pipe-nozzle had been
presented by either theoretical method [21] or numerical approach [22][23] in 1970s.
In the beginning o f the 1980s, Bryson, et. al. [24], True, et. al. [25], and Ranjan, et.
al [26] respectively, had presented a variety o f improved thermal stress analysis methods
on the pipe-nozzle. In 1986, Lapoint, et. al. [27], and in 1988, Baldur, et. al. [28], had
relative studies on thermal stresses o f the intersection area with [3 = 0.2, 0.4, 0.6, and y
= 5, 15, 25. Respectively, a reinforced nozzle on a cylinder due to thermal loads had been
studied with theoretical methods [30][31][32] and a numerical method [33], Also, some
applications o f thermal stresses analysis methods on cylinder-nozzle had been presented
[29][34][35], Strel'chenko, et. al. [36][37] had studied the temperature stress in T-shaped
intersection cylindrical shell with beta o f 0.2 and gamma o f 62.5 and 100, by means of
finite differential method (FDM).
In 1991, Moini, et. al. [38] discussed the specified boundary displacement method to
measure stress concentration due to geometrical discontinuity. Furuhashi, et. al. [39][40]
developed a simplified method o f stress analysis o f nozzle subjected to a thermal loading,
which can save costs and time in the calculation o f thermal stresses on nozzle-shell
connection. Their results was for a special geometry with [3 o f 0.254 and y o f 57.14.
-
From the above literature survey, it is obvious that normalized thermal stress factor
plots with extended range o f P and y values are necessary to facilitate the local thermal
stress computations o f pipe-nozzle.
The above publications and developments are chronologically tabulated as shown in
Figure 2.
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-
CHAPTER 3
BASIC THEORY
The purpose o f this study is to investigate the axisymmetrical thermal stresses around the
juncture o f a pipe-nozzle. Figure 1 shows a configuration o f this model.
3.1 Derivation of Equations for Deflections due to the Thermal Loading
Consider the thermoelastic state o f circular pipe, intersecting with a nozzle at right angle,
under the influence o f a steady state temperature gradient. The thickness o f the pipe may
vary according to any law but shall be symmetrical relative to the median surfaces, and
the ratio o f nozzle thickness / pipe thickness shall be unity. In deriving the basic relations
o f thin-shell theory, the Kirchhoff-Love hypotheses are used. The material from which the
shells are made is assumed to be homogeneous and isotropic. The thermal loading is
assumed to be such that geometrically linear thin-shell theory may be used. In addition, it
is assumed that the stress in the shells do not exceed the elastic limit o f the material. In
view o f the linearity o f this study, the overall stress at the pipe-nozzle juncture is
expressed as the sum of the stress states arising under the action o f a steady temperature
field. When a steady axisymmetric temperature field acts on the piping -nozzle, the forces
and the displacement o f the basic state are determined by solving the thermoelastic
problem [36][43] for each o f the shells.
12
-
13
The resolving equations o f the thermoelastic problem for the pipe shell and nozzle shell
with a linear temperature distribution over the thickness shall be
I'nix) = Tnm(x) + — y Tl](j{x) (])
Tp(z) = Tpm(z) + j — ^ T ^ z ) (2 )
where Tm = ( To + T, ) / 2 is the mean temperature o f a normal element o f the shell; Td =
( T() - Tt ) / 2 is the normal temperature difference; T„ and T, are the temperatures at the
external ( h = tn /2, H = tp /2 ) and internal ( h = -tn /2, H = - tp /2 ) shell surface. Refer to
Figure 1. Take the displacement forms
d 4w, 2 dD n\ \ d i w ( l d 2D „ \ \d 2w, R l(C „\\C „22~C „\2)dz4 D n u dz dz3 D„u dz2 dz2 D„uC,,u W1
R 3n (C n ] lC n2 2 -C n\2) ^ 2 a , R 3 f / d 2D n\\ , d 2D„\2s ,2t n C ‘ n tfl r-i ,3 1' -3 2 a 2 ' wL-'rtX 11- nl 1 u n\\ tn OZ* OZ*
^ , / 5 D n\\ d D n\2 dtn , I r\ r\ \ r o / ^ \ 2 O t n t i ) t / -> \2 ( - a T ' + _ &_ ) aF // ,+ (D / / i i + D //i2 )[2
-
14
_ \iEt„(x) , \iEtP(2) ,c ,n \2 j_ji 2 ’ ( p \2 j —1^2 (5c'
\iEt„ix) , _ J J L V % ( £ ) _" 12 1 2 ( l-f .i2) ’ P n 1 2 ( l-u 2)
At the ends o f the shells, the transverse forces and moments are zero.
The solution o f Eqs. (3) and (4) relative to wt and Wt allows the components o f the
basic state o f the shells under the action o f the axisymmetric temperature to be determined
from the formula
u t = a,Rn(T /im )(z ~ z Q) ; Ut = a (R p Tpm (6)
_ Dn\\ d 2w, 2 a , ( D „ u + D » i 2 ) mm x t - 0 2 n„ 2 1 nd ’
_ LSn 1 2 v ■ V ^ 12 1 n i l ) ,,,my t i>2 2 / ' W (7 a)
R l dz2 tn(z)
D n 12 d 2w, 2 a , ( n nn+D„22)r
K d z 2 t n(z)
Dp\\ d 2W, 2 a t ( D p n + D p\2)
R l d x 2 h
Dp\2 d 2W t 2 a t{ D p\2+DP22),
R 2P d x 2 h
a/ * =
^ p l2 C rjr , ^ix,v^p|2-r/vp22;,r , rp V .
Finding the forces and moments entails determining the functions o f Tp and Td
which are found from the resolving heat-conduction equations for the pipe,
d 2 I pm 1 d !pm Rp Rp 2 , Y.d y 2 y d y ip (K e + K i) rP™ (p ~ Ki ~ f pU
R 2P / 'i
-
15
d2Tpd i dl'pj r 12R 2 3R 2P/ 3R 2P/+ y ~ d f ~ + ~ (K
-
16
(I2 , 2R„ ,,, A .— + ~ i 7 l »
are satisfied.
The components characterizing the thermal stresses o f nozzle are found using the
homogeneous system o f equilibrium equations
dnx i ^ _ n . d,,y t o y .dz +
■ŷ 1 + +mt - M
-
17
nxy = C /,3 3 ^ 1 i mx = ~ iP n \ \ + D n \ 2 xsn2>
my = '~(D n22V}n 2 + D n l 2 rDn O > mxy = _2Z)aj33t /;1 ( 14a)
+ Cp \2 ^ p 2 ’ % = Cp 2 2 C=p2 + Cp \ 2 ^ p \
Ny (P = C/?33^/?l ’ % = “ ( ^ p l 1 wp \ +Dp l 2 wp2)
M » n 3 3 ~ 2 ^ ( 15b>
Etp . n _ |lEtp n _ Etp+ P
*-.* A73D/>1 1 = Dp22 = I 2 ( l - p 2) ’ D/>12 = 72(1- p 2) ’ D/>33 = 24(1+10 (15d)
C/>11 Cp22 !-(x2 * CP X2 1 -p 2 ’ Cp33 2 ( 1 -4 1 ) (15C)
AY3 p£Y3 E llP • r> P • r> _ P
The geometric equations o f deformation and change in curvature for the pipe and
nozzle are as followings
c „ i = £ | ; = £ < ! + ! >
_ 1 d2w . _ ___ 1 d2w . _ 1 d2w n A vf?2 dr 2 ’ " 2 i?2 d
-
18
1 d l l sin2cp^ ^ i r ^ + - K T w c»? =
1 dVP 2 y R n dip n
^ /^w dy y R n dip yM? /fy?
1 d2ET ra/;1 / « ay2
= —r W
P2 y 2R l dip2 y R l d
-
19
y = 2 / /;0)sin//
-
20
After substituting Eqs. (17) into the resolving equations for the pipe and nozzle, the
method o f variable separation is undertaken. As a result, two new systems o f
differential equations with variable coefficients are obtained. Variable separation is also
undertaken in the matching boundary condition o f the pipe and nozzle.
3.2 T herm al Stress Factors
In order to present different configurations o f the pipe-nozzle model, all the thermal
stresses are represented in dimensionless form as thermal stress factors. For thin shells, the
assumption o f a linear temperature gradient through the thickness is a good approximation
so that the temperature distribution will become
T = Tavg + ~ ~ d t (18)
where Tavg is the average wall temperature and A T is the difference between the outside
and the inside wall temperature; t is the thickness o f pipe or nozzle and there is a point at
distance dt from the median surface in th meridional direction. From Timoshenko's Theory
o f Elasticity [43 ], the thermal membrane force, N,,, and thermal moment, are
Etarfavg E t2
-
21
K T ’ ’̂ r ' rs'r
By the finite element method, all the thermal stress factors on the longitudinal and
circumferential directions and sign notations for thermal stresses on the pipe o f
pipe-nozzle are added into a summarized Table 1 ( Ref. Fig. 4 ) which had been presented
by WRC 107 [2], Data presented in various plots are shown in the Appendix A from
Figure IT to 8T for the longitudinal and circumferential thermal stress factors on the pipe,
and Figure from 9T to 16T for the longitudinal and circumferential thermal stress factors
on the nozzle. Numerical examples o f thermal stresses on the pipe and the nozzle are given
in Chapter 6.
-
22
Table 1 Modified stress computation table o f WRC 107 including local thermal stresses
F ormF ig .
Read Curves For
Stress Factor
Compute Absolute Values o f Stress & Enter Result
(psi)Dl
3C11' N ,P m , nlpm„ r , t '
1C“> + + +3 A " ’ AL
A/cAfliP)Knl A t. , A icM cW iV ) R lfiT
■mt-mmm + +
f l r r n1 A(,) A /,A/c/(B«p) **1
AT, Wc A /c /(» .p ),t t« p r 1 +
3B111 tv.AWfHiP) Knl
AT, A/t + + anmmIB or IB-1 M .
M lW*. P) Kb[M , , i t i i + +
5T-8TC ircum ferential
2(1-P)E ariT
E u t A T . 1(1 ~u) _ td-ai'farar ■ + + + +
Add algebraically summation of circumferential stresses, aUL
4C10 JVl.p/H .
« t tVj * _p _ _
2C(,) M xP
.. .A4xs6P _«■£! p Ifi + + + +
4A(1) N ,Afct(« iP)
K 1 I°VMcWltV)]Rl$T '2A(I) Mx
McKXm P)jt' I A /r . 6Afc
‘ 'iW c /f /t.p i 't i .p r1 '
4B1" NxAfrWiP)
(W**«5N!o»«
-
23
Z
LL
P = Radial Load, lb.Vc = Circumferential Shear, lb.VL = Longitudinal Shear, lb.To= Outside Temperature, °F T,= Inside Temperature, °F
Mc = Circumferential Moment, lb.-in Ml = Longitudinal Moment, lb.-in. Mj.= Torsional Moment, lb.-in.Mth= Thermal Moment, lb.-in.
F igure 4 Different loadings applied on the juncture o fpiping-nozzle ( ref. to Table-1, computation and sign notation sheet for local stresses o f piping- nozzle)
-
CHAPTER 5
FIN IT E ELEM ENT M O D EL
4.1 G eneral
Because there is no suitable mathematical model and exact solution available in simulating
the real pipe-nozzle geometry, a finite element analysis ( FEA ) has been utilized in this
thesis. It is understood that the finite element analysis method with computer simulation
has provided an increasingly important role in engineering design and analysis. It also
performs speedy and reliable calculations and develops a comprehensive, accurate, and
efficient procedure for local thermal stress analysis at the juncture o f pipe-nozzle.
However, the varying sizes o f the pipe-nozzle at the juncture, cause difficulties in
obtaining accurate and economical solutions by the finite element method. Therefore, it is
extremely important to develop the proper number o f nodes and generate sufficient
meshes to provide a efficient finite element model.
In this thesis, ten full size finite element models, each with a specific beta value, were
developed. Each model with approximately 5000 node points and 3000 elements, were
generated by the ALGOR finite element program with "Superdraw" computer code.
[41][42], All the computations were performed on a 486/DX-66 personal computer with 8
M ega RAM and 300 Mega Bytes Harddrive memory. It took about 10,000 seconds o f
CUP running time for each computation.
24
-
4.2 A ssum ption
For the analysis, the following assumptions were applied:
1. The material is assumed to be homogeneous and isotropic, and obeys Hook's
law. The resulting stresses are within the proportional limit o f the material.
2. The influences o f self-weight are neglected.
3. The internal pressure is the same as ambient pressure.
4. There are no transitions, fillets, or reinforcing pad at the junction.
5. The steady state temperature distribution is linear and the inside temperature is
higher than the outside temperature.
4.3 Asymptotic Studies
For optimum accuracy and convergence within the framework o f the program, the finite
element model o f quadrilateral thin shell is adopted. Two important asymptotic studies
w ere introduced:
4.3.1 Asymptotic Study of Node Points at Juncture of Pipe-nozzle
Figures B1 to B 16 in Appendix B showed the convergence o f various thermal stresses at
point A and C ( Figure 1 ). As the number o f node points on the pipe-nozzle juncture
model increased to 96, all the thermal stresses converged asymptotically. In this case, the
density o f mesh on the juncture o f pipe-nozzle satisfied the asymptotic requirement to
avoid any influence o f the mesh element to the thermal stress values.
-
26
4.3.2 Asymptotic Study of the a p and a n
As for the influence o f boundary parameters, a p ( pipe length / pipe mean radius ) and a n (
nozzle length / nozzle mean radius ), to the solution o f various thermal stresses, Figures
C l - C16 in Appendix C showed the percentage o f improvement with larger a p to the
previous a p and Figures C17-C32 showed the percentage o f improvement with larger a n
to the previous a n . It is evident that a p = 8 and a n = 4 are the optimum quantities
that boundary conditions would not have any significal effect on the outcome o f the
thermal stresses at the pipe-nozzle juncture.
4.4 Normalization studies
Normalization studies are made to verify the validity o f using geometrical parameters,
beta, P (nozzle mean radius / pipe mean Radius) and gamma, y (pipe mean radius / pipe
thickness) to express the local thermal stresses . There are four different cases discussed
as followings and numerical data are listed in Appendix D.
4.4.1 Case I, II, III
Case I assumed that a p = 8, a n = 4, P = 0.6, and y = 50. By using two distinct geometries,
both having the same geometric parameters , i.e. a , P, and y , and ST, Table D -l to D-3
in Appendix D showed that both models have identical local thermal stress results when
model #2 is twice the size o f the model #1. This verified the validity o f using a p , a n , P,
and y as geometrical parameters to express the local thermal stresses.
Case II had model #3 and #4 w ith a p = 8, a n = 4, P = 0.3, and y = 100, the local
-
27
thermal stresses are listed in Table D-4 to D-6, respectively.
Case III had model #5 and #6 with a p = 8, a n = 4, {3 = 0.9, and y = 20, the local
thermal stresses were listed in Table D-7 to D-9 respectively.
Again, from Table D-4, D-5, D-6 and D-7, D-8, D-9, the geometric parameters o f
a p, a n , P, and y were valid.
4.4.2 Case IV
Case IV had tw o models (#7 & #8) which showed that the normalization o f thermal
2xcyx(I-|i)stresses with stress factors ------ — are valid when the temperature for each model wasCxatxAj r
assigned 400 °F and 900 °F, respectively. Table D- I0 to D-12 tabulates the local thermal
stresses and stress factors.
-
CHAPTER 5
C O M PA RISO N O F DATA
5.1 G eneral
For the thermal loading, the thermal stress factors induced by the steady state thermal
gradient are compared with the related literatures cited in Chapter 2. Basically, there is no
sufficient numerical data that can be used for comparison purposes.
5.2 C om parison of T herm al Stress Factors
There are two cases being discussed as followings:
5.2.1 Case 1
F. Cesari [18], presented a model with a pipe radius = 191.4 mm, pipe thickness =
6.7 mm and nozzle radius = 24.675 mm, nozzle thickness =1.350 mm, where the value of
beta, [3 is equal to 0.129 as well as gamma, y is equal to 28.57. The structure was
subdivided into 96 elements with 66 node points and there were only 16 node points at the
juncture o f the pipe-nozzle. The temperature difference between internal and external
pipe-nozzle was 25 0 C. The Young's Modulus, the coefficient o f thermal expansion
and the Poisson's ration were given as 1.7 x 105 N /m nr , 1.85 x 10's mm/mm °C , and 0.3,
respectively. The maximum thermal stress found at node point C ( Figure 1 ) was 406
N /m nr . In this manner, the thermal stress factor, KT , at node point C can be calculated as
the following :
28
-
29
i - i 1 ■ ■i' OS
C«VI
-
30
o r 7 ’ x 2 x ( 1 - L i ) 4 0 6 x 2 x ( 1 - 0 . 3 )
K -------------- — = ---------------- '-------- — 1 23T E x a r x A T 1.7x 105xl ,85x 10~5x25
from Figure T3 in Appendix A, with 3 = 0.129, and y = 28.75, the local thermal stress
factor should be approximately equal to 5.5.
Compared these two thermal stress factors, there is a percentage o f derivation o f
23%. One may detect that Cesari's results did not have sufficient node points or meshes to
ensure the accuracy o f the results. Additionally, his paper did not take into consideration
o f the boundary condition o f the pipe as well as nozzle. Figure 5 shown the comparison of
thermal stress factor between FEM data and Cesari's data.
5.2.2 Case 2
A paper presented by A. S. Strel'chenko, et. al [36], had a model with the pipe radius =
0.25 m, pipe thickness = 0.0125 m and nozzle radius = 0.05 m, nozzle thickness = 0.0025
m, where the value o f beta, 3 is equal to 0.2 and gamma, y is equal to 20. In this study, a
numerical finite-difference method (FDM) was employed to solve the differential
equations and FORTRAN IV program was developed to calculate the stress values. The
temperature difference between internal and external of the pipe-nozzle was 30 °K. The
Young's Modulus, the coefficient o f linear thermal expansion, and Poisson's ration were
given as 205.8 GPa, 1 x 10'5 m/m-°K , and 0.28, respectively. The thermal stresses and
c t t x 2 x ( 1 - h )
thermal stress factors were given in Table 2 based on the equation KT= •
-
31
Table 2 List o f thermal stresses and thermal stress factors given in the case 2by A.S Strel'chenko [36] and by FEA data o f this thesis
Thermal Stresses, MPa
in the longitudinal direction at node point o f the nozzle
A0 A, C0 c,
by A.S. Strel'chenko [36] 129.3 108.1 35.4 17.6
by FEA Data 145 109 120 111
in the circumferential direction at node point o f the nozzle
A0 A, C0 c,
by A.S. Strel'chenko [36] 91.8 25.1 25.1 39.8by FEA Data 190 192 214 175
Thermal Stress Factors
in the longitudinal direction at node point o f the nozzle
A0 A, Co c,
by A.S. Strel'chenko [36] 3.02 2.52 0.86 0.41
by FEA Data 3.4 2.6 2.8 2.6
in the circumferential direction at node point o f the nozzle
A0 A C0 c,
by A.S. Strel'chenko [36] 3.02 0.59 0.59 0.93
by FEA Data 4.4 4.5 5 4.1
As a result o f comparison for these thermal stress factors showed in figures E l to E8
in Appendix E, there exists a minimum percentage o f derivation o f approximately 3.2%.
Also FEA data are all greater than that o f Strerchenko's data, which imply that the FEA
results are much more conservative. However, in Strel'chenko's paper, the length o f pipe
and nozzle were not reported, it may explain the discrepancy o f the results.
-
32
5.3 Comparison of Thermal Stresses with a long hollow cylinder
A comparison o f local thermal stresses at the juncture o f pipe-nozzle and theoretical
thermal stresses at the regular long hollow pipe is made as in the following:
The maximum theoretical thermal stress at the regular long hollow pipe without
nozzle can be calculated by using equations (23) and (24) [44]
a t a = ° z a = ^2 3) fo r the inside su rface ’
a tb = a zb ~ ” (24) for the outside surface
where subscript t refers to the circumferential direction and the z refers to the axial
direction o f the pipe.
Comparison o f the local thermal stresses at pipe-nozzle and the theoretical thermal
stresses at a regular long hollow pipe are tabulated in Table 3, which one may observe
that,
1. The longitudinal local thermal stresses at node points Au , AL , Bu , and BL on the
pipe region o f pipe-nozzle are smaller than the theoretical thermal stresses on a regular
long hollow cylinder. On the contrary, the longitudinal local thermal stresses at C y , CL ,
D jj , and D L on the pipe region o f pipe-nozzle are greater than the theoretical thermal
stresses on a regular long hollow cylinder.
2. Both longitudinal and circumferential local thermal stresses on the outside surface
o f the pipe o f pipe-nozzle are greater than the theoretical thermal stresses on a regular
long hollow cylinder and on the inside surface, neither local thermal stress is greater than
the theoretical thermal stress on the regular cylinder.
-
33
3. All the local thermal stresses on the nozzle are greater than the theoretical thermal
stresses on the regular long hollow cylinder.
Table 3 Comparison o f local thermal stresses at pipe-nozzle _________ and theoretical thermal stresses at regular long hollow pipe
Longitudinal Direction Au a l Bu BL Cu CL Du DlLocal thermal stresses on pipe
region o f pipe-nozzle25630 -21730 25630 -21730 65185 -61285 65185 -61285
Theoretical thermal stresses on regular long hollow pipe
55714 -55.714 55714 -55.714 55714 -55,714 55714 -55,714
Circumferential Direction Au Al Bu BL Cu CL Du DlLocal thermal stresses on pipe
region o f pipe-nozzle71315 -48470 71315 -48470 27300 -22840 27300 -22840
Theoretical thermal stresses on regular long hollow pipe
55714 -55,714 55714 -55,714 55714 -55,714 55714 -55,714
Longitudinal Direction A0 A, Bc B, C 0 C, D 0 D,Local thermal stresses on nozzle
region o f pipe-nozzle111420 -42342 111420 -42340 83565 -68245 83565 -68245
Theoretical thermal stresses on regular long hollow pipe
55714 -55,714 55714 -55.714 55714 -55.714 55714 -55.714
Circumferential Direction A0 A, B 0 B, C0 C, D 0 D,Local thermal stresses on nozzle
region o f pipe-nozzle125348 -118384 125348 -1 18384 115600 -114206 J 15600 -114206
Theoretical thermal stresses on regular long hollow pipe
55714 -55,714 55714 -55,714 55714 -55,714 55714 -55,714
-
CHAPTER6
NUMERICAL EXAMPLES
To calculate the local stresses on the pipe o f pipe-nozzle due to external loadings with
steady state thermal gradient, an example is given as in the following :
6.1 Example I
A 12.75 inch O.D. pipe is intersected by a 5.325 inch diameter nozzle. Both pipe and
nozzle thickness are 0.375 inch. The pipe mean radius, , can be caucluated as (pipe
O.D. - pipe thickness) 1 2 = (12.75 - 0.375) 1 2 = 6.188 inch, as well as nozzle mean
radius, rm , is equal to (5.325 - 0.375) 12 = 2.475 inch.. As a result, beta, P = x j R,,, = 0.4
and gamma, y = R,,,/ tp = 16.5 . However, alphap, a p (Pipe length / pipe mean radius) is
equal to 8 and alphan, a n (Nozzle length / nozzle mean radius) is equal to 4 , in accordance
with the previous discussion. A 500 °F internal temperature and 100 °F environmental
temperature are assumed in this example and the material o f both pipe and nozzle are 347
stainless steel. The material properties o f this pipe-nozzle model are listed in Table 4.
Table 5 shown its geometrical parameters and its dimensions.
The thermal stresses were calculated by taking the dimensionless thermal stress
factors ( Kt ) from Appendix A, which are also listed in Table 6, and then multipling it
.£fX(X y yy rf*with ,1— r- • Table 7 is the modified stress computation table from WRC 107 which
2 x ( l -p )
taken into account the local stresses on the pipe o f the pipe-nozzle due to external loading,
as well as local thermal stresses.
34
-
For the external loadings (refer to Figure 4), it assumed that
Radial Load, p = 400 lb. ( downward )
Circumferential Moment, Mp = 500 lb.-in.
Longitudinal Moment, ML = 500 lb.-in.
Torsional Moment, MT = 500 lb.-in.
Circumferential Shear Force, Vc = 300 lb.
Longitudinal Shear Force, VL = -400 lb. ( to the rig h t)
Table 4 Material properties o f the illustrating pipe-nozzle model
ctj. = Thermal Expansion Coefficience 6.50e-06 in/in-°F
E = Young's Modulus 3.00e+07 psi
p, = Poisson's ratio 0.3
Tj = Internal Temperature 500 °F
T0 = Environmental Temperature 100 °F
5T = Tj - Tc 400 °F
Pipe Material 347 SS
Nozzle Material 347 SS
Table 5 Geometrical parameters and dimensions o f example forcalculation o f local stresses on pipe o f pipe-nozzle model
a D = Pipe length / pipe mean radius 8
a n = Noz. length / noz. mean radius 4
P = Noz. mean rad. / pipe mean rad. 0.4
y = Pipe mean rad. / pipe thk. 16.5
Lp = Pipe length 49.5 ins
Ln = Nozzle length 9.9 ins
Rm = Pipe mean radius 6.188 ins
r = Nozzle mean radiusm 2.475 ins
tD - Pipe thickness 0.375 ins
tn = Nozzle thickness 0.375 ins
-
36
In Table 6, the longitudinal thermal stress factors at node point Ay and Ay are read
from Figure IT and 2T o f the Appendix A, as wll as Cy and CL from Figure 3T and 4T.
Because o f the axisymmetry on the pipe-nozzle geom etry, numerical value at node point
By should be identical as the value at Ay . Similarly, for BL is equal to AL , Dy is equal to
Cy , and DL is equal to CL. With regard to circumferential thermal stress factors, they are
from Figures 5T to 8T o f the Appendix A.
T ab le 6 Com putation table o f therm al stress factors on pipe
Read values from thermal stress factor plots
Au a l Bu b l Cu C L Du d l
in the circumferential direction from figure 5T-
8T1.28 -0.87 1.28 -0.87 0.49 -0.41 0.49 -0.41
in the longitudinal direction from figure 1T-
4T0.46 -0.39 0.46 -0.39 1.17 -1.1 1.17 -1.1
Tw o calculations are given as followings to illustrate how the therm al stresses can be
obtained :
The therm al stress factor in the circumferential direction at node point Ay (Figure 1),
which can be found from Figure 5T in the Appendix A, is equal to 1.28. Therefore, the
thermal stress is able to be com puted by the formula
K t x E xcltxA T 1.28x3.0x107x6 .5x10-6x400 ._ 2“ ( l ^ ) = -------------- 2 x (l-0 .3 )------------- " ?1’315 pS‘
By the same way, the thermal stress factor in the longitudinal direction at Ay is given
from Figure IT as 0.46 and the thermal stress should be reckoned as
K tx -E x c ltx A T 0 .46x3.0x107x6 .5x10-6x400 1 c r in 2 x ( l - n ) = ------------- 2 ^ 1 = 0 3 ) ------------- = 2 5 6 3 0 PS'
-
37
Table 7 Modified stress computation table o f WRC 107 including local thermal stresses on the pipe- numerical example
FormFig.
Read Curves for
Stress Factor
Compute Absolute Values of Stress & Enter Result
(psi)A« A , Bu Cu Cu D u
3C(‘> - ^ 1 2 2PIK.F , At* . p"'pirJ r.t
•210 -210 -210 -210 -210 -210 -210 -210
1C(I) = 0.0423 .. P _* 6* p ^ -723 723 -723 723 -723 723 -723 723
3A"' - I 237A /c W iP )
y \ i A/cn ,A M * iP ) R i p r
l i l i
, .in . i f ™ !
-62 -62 62 62
1A(,» 0130A M * . P)
y , AT¥ 0A/C* l A W (fi,P )lR , p r J
P i l l -647 647 647 -647
3B“>W ^ P ) = 2
-
38
After all the calculations o f thermal stresses been done, the data can be input into
Table 7 to complement the WRC 107 computation table and obtain the maximum stress
intensity. However, these local stresses are only applied on the pipe portion. For the local
thermal stresses on the nozzle, should be combined with local stresses due to external
loadings, which presented by Lin [45],
6.2 Exam ple II
A second example is given here to compute the local stresses on the nozzle region of
pipe-nozzle due to external loadings with steady state thermal gradient:
A pipe with O.D. o f 100.25 inch, thickness, t o f 0.25 inch, and a nozzle with O.D.
o f 12.75 inch, also has a thickness, tn of 0.25 inch. The pipe mean radius, Rm is equal to
50 inch and the nozzle mean radius, rm is equal to 6.25 inch.
Therefore, beta can be obtained as P = ^ - = 0.125 , and gamma as y = 7 1 = 200 .•*m 'p
The same external loadings and material properties as the example one were using in
this calculation. The geometrical parameters and dimensions o f this example is shown in
Table 8 .
Again, the thermal stress factors o f the nozzle are calculated by taking the
dimensionless thermal stress factor from Figures 9T to 16T o f the Appendix A and they
are shown in Table 9. Table 10 is the modified stress computation table with thermal
stresses on the nozzle region o f the pipe-nozzle.
-
39
Table 8 Geometrical parameters and dimensions o f example for _________calculation o f local stresses on nozzle o f pipe-nozzle modela p = Pipe length / pipe mean radius 8
a n = Noz. length / noz. mean radius 4
(B = Noz. mean rad. / pipe mean rad. 0.125
Y = Pipe mean rad. / pipe thk. 200
Lp = Pipe length 400 ins
Ln = Nozzle length 24.9 ins
Rp, = Pipe mean radius 50 ins
r = Nozzle mean radiusm 6.25 ins
tp = Pipe thickness 0.25 ins
tn = Nozzle thickness 0.25 ins
Table 9 Computation table o f the thermal stress factor on nozzle
Read values from thermal stress factor plots
A0 A, B0 B, c 0 C, Do D,
in the circumferential direction from figure 13T
to I6T2.25 -1.125 2.25 -1.125 2.075 -2.05 2.075 -2.05
in the longitudinal direction from figure 9T
to 12T2.0 -0.76 2.0 -0.76 1.5 -0.76 1.5 -0.76
Table 11 lists the local thermal stress factors due to different external loadings from
Lin [45],
-
40
Table 10 Computation table o f example for calculation o f local stresses on nozzle o f pipe-nozzle model
FormFig.
Read Curves for
Stress Factor
Compute Absolute Values o f Stress & Enter Result
(psi)A, D,
1 IP & 15P1" = T a i l t - I I " lP m . HmT "-154 -154 -49
9 P & 13P"> = Table- II m p • j 'l-1,920 1,920 -1,920
7M C(l) = T ab le - I I Kt\[N 9 M c
S «
5MC(l> m 9M d {R m \»)
A / ,
MdVlm&VRmVr16 M c H -553 553 -553
7ML(1) K t . Mi"WwiPrJiipr*112
5ML(,) A / .MMmV) = T ab le - I IK « . 6Ml _
b lM ,f(R m\D lR mp T 2 '
13T-16TCircumferential
2(l-|i)Eaj&T- table - 9 Ittrdr̂ tl-lO = 2(i-^J£tt7Ar 125348 -118384 125348 -118384 115,600 •114,206 115,600 ■114,206
Add algebraically summation o f circumferential stresses, a =124,536 -118,103 124,931 •118,051 113,058 •111,802 114,205 •112,867
12P& 16P0* N xPIRm
= T a b le - I t Ka\ N x , P P /R m l R * T ~
-18 -18 •870 -870 -870
I0P&14P°} ^ = Table - 11 tr rAf* 16P _Kbl'p-lp ~ -1306
8MC(1) N xiWcVfflip) = Table - 11 K f I 4 / colA/c/(«SP) VtiPT*'■mm 123 123
6M Cll) M r tr r A / x 1 6 M c *blMd(RmPVRmpT2 ' -184
8ML“ > N x MJ(RlP) ' KalN x , M i =
WtRipj'flipr iHH
M S6ML0) Mx = Table - 11 *Al
Mx i 6Afi 258 258
9T-12TLongitudinal
2(1-10 _ £ar&r "
gaMr.2(l-M) . 2(l-)0J£ar&r '
111,420 -42340 111,420 -42340 83,565 ♦68,245 83,565 -68,245
Add algebraically summation o f longitudinal stresses, a =109,822 -40,811 113,151 -41,294 81,83 -68,555 82,503 -68,678
Shear stress due to the Torsion, MT
M t
Shear stress due to the Load, Vc txtp =■mm- i f
Shear stress due to the load, VL Txcp = V i*rmT ~
imm
Add Algebraically for Summation o f Shear Stresses, x•25
-
41
T able 11 Local stress factors on the nozzle from Lin, Sun, and Kopiik [45]
Read values from Stress Factor Plots
A0 A, B0 B, c 0 c , D0 D,
From Figure 1 IP 0.895 0.895 0.895 0.895 Ifllllsll f * ** Z V
From Figure 15P: ' 's >\
< ■»iiVf-y - ......
0.285 0.285 0.285 0.285
From Figure 9P 0.027 0.027 0.027 0.027s4t̂. f&teyssssssife:XwV'/.-.XvS 0.112 0.112 0.112 0.112
From Figure 7MC % A „ 3 ^ -t 0.4 0.4 0.4 0.4
From Figure 5MC i l l s ® * * < >*• > 0.111 0.111 0.111 0.111
From Figure 7ML 2.24 2.24 2.24 2.24 * JWWMvSrM̂ K'M\l̂Vy •i, >* *«•* ;
From Figure 5ML 0.017 0.017 0.017 0.017 l l§ § l ljŝS-S
V, r,< j-:
From Figure 12P 0.104 0.104 0.104 0.104 ... vS.< :
From Figure 16PsSTO&M
5.046 5.046 5.046 5.046
From Figure 1 OP 0.078 0.078 0.078 0.078 ilWWWWW- 'U,< -1 Si*.?-,From Figure 14P > < * f/ X < < JX ; >+> <
i i 'u n r l l l i m
n \A\ * : t s 4 > * ' V 0.03 0.03 0.03 0.03
From Figure 8MCX- \ «V: < * y V «*■ •fi-vex v>4 y *< 2.456 2.456 2.456 2.456
From Figure 6MC F i f: v -«v» i. -tv > 1 V * v h ,-K *.J a / > <
% * 0.037 0.037 0.037 0 037
From Figure 8ML 0.319 0.319 0.319 0.319 ||$ $ ^ || ' * * ; vFrom Figure 6ML 0.052 0.052 0.052 0.052
..........
|§ it|p l; * > *>■ V
' yWiW^fcSwS?
-
CHAPTER 7
CO NCLU SIO NS
Since the finite element techniques is capable o f simulating the true geometry o f the
pipe-nozzle configuration, node points and boundary condition studies prior to production
runs ensure that the local thermal stress factors presented in this thesis are accurate and
reliable. These local thermal stress factor plots are shown in Figures IT to 16T of
Appendix A. Again, these local thermal stress factors may be used in conjunction with
WRC 107 with other external loadings.
By studying the Figures IT to 16T o f Appendix A, the following conclusions may be
made:
1. When the gamma value increases, all the thermal stress factor values are
increasing, i.e., the thinner the shell, the higher the local thermal stress.
2. At the node points A and B o f the pipe, the local longitudinal thermal stresses are
always less than the local circumferential thermal stresses, on the contrary, at the node
points C and D o f the pipe, the local longitudinal thermal stresses are always greater than
the local circumferential thermal stresses.
3. On the nozzle, the local circumferential thermal stresses are always greater than
the local longitudinal thermal stresses.
42
-
APPENDIX A
THERMAL STRESS FACTOR PLOTS
43
-
44
o o o o o o m o cn cm r-i
m o in m m on r-> —,
f ♦i :
ooo ’
o
m©
r f©
enO\ \ \
-
45
o o o o o o in om (N
o u~> >n »n or - ro n r - r i
! i i H 1T *
OO
CNo
o
oqo
r-;©
vqo
•qO
^ro ’
-
46
o o ° o ' n 0 ' 0 ' n , n 0m r-c
C "- V I C O ( N
©
d
00
o
©
o
vo in ■'* COCNCO
iojob^ ssaijg [Buu^qx
Figu
re
3T Th
erm
al s
tress
fa
ctor
in
the
long
itudi
nal
dire
ctio
n at
poin
t C
v of
the
pipe
-
47
g g ®
i l lI !
i " 0 m tr(N CSr oO
O SO
°oo
c-d
VOo
mo
^rd
C*1d
oio
03-
i o p u j ssa jjg lBUU9lLL
Figu
re
4T Th
erm
al s
tress
fa
ctor
in
the
long
itudi
nal
dire
ctio
n at
poin
t C,
of
the
pipe
-
o vn
* ‘ \ * ® * *
sso i’VS ^
-
49
f n t N n r , h ln r ,1 ( N "
' - ■ i i ' :{ I i { ♦ b 4 ii ! I I ! I : I
o
ood
md
rro
ONO t~- vo m v CN O
jojdbj ssajjg iBuuaqx
Figu
re
6T Th
erm
al s
tress
fa
ctor
in
the
circ
umfe
rent
ial
dire
ctio
n at
poin
t A,
of
the
pipe
-
50
o o o oO O V ^ O m CM r-H ^
m o in m in o» n r o cn t-h
■ i i* i * : I •
oOro cn O
OsO
00o
r-o
'
-
51
o o o o o o m oC O ( N 1—< * -1
in o m in m
II 4 i
T T ■ t T
jo jo b^ SS91JS I’G u u aq x
Figu
re
8T Th
erm
al s
tress
fa
ctor
in
the
circ
umfe
rent
ial
dire
ctio
n at
poin
t C,
of
the
pipe
-
52
o © o o o o m oW N r l r -
m o in in «n o in cn c n f-H
i )
O
00©
o
d
cnd
J 0 J 0 B J S S 9 JJS [B U U Q q X
Figu
re
9T Th
erm
al s
tress
fa
ctor
in
the
long
itudi
nal
dire
ctio
n at
poin
t A„
of
the
nozz
le
-
53
r ' in m n
i
IIII»I
liIi/
I1I4IIt
III4 ■II
I
IIii
rii
;i
j4i
i
t1
1Ii
i i
1
1
1
1 1
4
101
IbI
1
I \l l
1 I\
\ \ * Iz ■
t Vf t
¥
\ I• I
V •1
i
"
1 i I I/
«
9Ovo
00o
t"d
voo
QQ.«nd
d
md
CNd
CNo C"0 *1 Tf
JO JDBJ SSSJJg IBUIJSqjL
Figu
re
10T
Ther
mal
stre
ss
fact
or
in the
lo
ngitu
dina
l di
rect
ion
at po
int
A, o
f the
no
zzle
-
54
o o o oO O m O i n o m i n m o m ro cn - h —<
: I !
CNo r^o in r̂
iO JD B J SS9JJS {BUiiOqjL
Figu
re
11T
Ther
mal
stre
ss
fact
or
in the
lo
ngitu
dina
l di
rect
ion
at po
int
C„ o
f the
no
zzle
-
55
o o o o m in oM C l t O N f H H
I II •
II?I
$ *
_I
00o
©
ino
Tf
CN
o
o
tT CM r~
-
56
J O P B jJ SS9JJJJ [ B u u o q x
Figu
re
13T
Ther
mal
stre
ss
fact
or
in the
ci
rcum
fere
ntia
l di
rect
ion
at po
int
Ao o
f the
no
zzle
-
57
0 0 ^ 0 1 0 0 'n 'n , n 0 M r i n
■ II
l ♦i ■
1 I ■* 4 *
I !
o
oo
mo
CNo
o
CNm m
-
58
J0P ^ ss3 J1 S l' UU94'L
-
59
4
0J>NOC
-
APPENDIX B
ASYMPOTIC STUDY OF NODE POINTS
AT THE JUNCTURE OF PIPE-NOZZLE
60
-
0.70
0
61
OSCN
CN
mCN
-
0.70
0
62
CN
c nCN
U*NN0 C1
-
1.0
00
63
m
osCN
(N
ON0 c1w-coS
O 'COco
oa .
-
1.10
0
64
OSc n
c n
vsCN
cnCN
N0 a1uD.
oUiCO3
aCOco
OCL
-
1.0
00
65
m
o sCN
CN
-
oo r i
66
O sCN
C-"CN
inCN
c nCN
JJNN0 c1ua
oca3
o
TDOz«+-
-
0.75
0
67
o sCN
t>C N
IDC N
c nC N
NNoc t
-
0.70
0
68
osCN
CN
cnCN
CN
o
NN0 c1
TJO
Z
oZ
io io bj sssijg jBunaqx
Figu
re
B8
Con
verg
ence
of
node
po
ints
at the
ju
nctu
re
of pi
pe-n
ozzl
efo
r ci
rcum
fere
ntia
l th
erm
al s
tress
fa
ctor
at
poin
t C,
of
the
pipe
-
1.15
0
69
m
OSCN
r~~CN
CN
c nCN
JUNNOCi
-
1.20
0
lop B jj ssa ijs [BUJisqx
Figu
re
BIO
Con
verg
ence
of
node
po
ints
at the
ju
nctu
re
of pi
pe-n
ozzl
efo
r lo
ngitu
dina
l th
erm
al s
tress
fa
ctor
at
poin
t A
of the
no
zzle
-
1.23
0
71
Os 00 r " SO in T f cn CN oCN CN CN CN CN CN CN CN CN CNCN CN CN CN CN CN CN CN CN CN
>-« r - t i T-^ T—< i—i
xoiobj s sa n s iB u iisq x
Figu
re
B11
Con
verg
ence
of
node
po
ints
at the
jun
ctur
e of
pipe
-noz
zle
for
long
itudi
nal
ther
mal
stre
ss
fact
or
at po
int
C, o
f the
no
zzle
-
ooei
72
OsCN
>T)tN
coCN
CN
-
1.50
0
73
c n
CM
—
N0 C143D,
03u*CO3
aeaco
oOh03-aoZ
oZ
o o O O O O o O o oin o ID O in o m O in orr Tt; cn cn CM CM © oV -H F « * r “ » r—■ _
io ôbj; sssjjg jBuijsqx
Figu
re
B13
Con
verg
ence
of
node
po
ints
at the
ju
nctu
re
of pi
pe-n
ozzl
e fo
rci
rcum
fere
ntia
l th
erm
al s
tress
fa
ctor
at
poin
t A
of the
no
zzle
-
1.60
0
74
m
OsCN
r-CN
mCN
JUNN0 C1
a .
-
1.80
0
75
m
CN
inCN
joNN0c1
-
008*1
76
cn
CN
CN
cnCN
J3' nN0 c14)CU
-
APPENDIX C
A SY M PO TIC STUDY O F CXp AND a n
77
-
78
-
79
eR
o
00
\o
oesoo
a
JU 9lU 9A 0dlU J SgB JU aO ISd
Figu
re
C2
Perc
enta
ge
of im
prov
emen
t fo
r la
rger
a
p to
prev
ious
a
( of
long
itudi
nal
ther
mal
stre
ss
fact
or
at A,
of
the
pi
pe
-
80
o
00
od
(Nd
cs CO
juau iS A oduJi aS B ju ao iad
Figu
re
C3
Perc
enta
ge
of im
prov
emen
t fo
r la
rger
ap
to pr
evio
us
ap
of lo
ngitu
dina
l th
erm
al s
tress
fa
ctor
at
C1:
of the
pi
pe
-
81
o
oo
o od
CN CMo
oo
JU 9U 13A 0duiI sSBJUaOJQcI
Figu
re
C4
Perc
enta
ge
of im
prov
emen
t fo
r la
rger
a
p to
prev
ious
a
p of
long
itudi
nal
ther
mal
stre
ss
fact
or
at C,
of
the
pipe
-
82
CN
O
OO
o'Oo
o CNo
CN OOd
ju9iu9Aoduii aSBjuaojad
Figu
re
C5
Perc
enta
ge
of im
prov
emen
t fo
r la
rger
a
p to
prev
ious
a
p of
circ
umfe
rent
ial
then
nal
stres
s fa
ctor
at